Copying and pasting comments in Google sheets does not work at the present time. However, after some research I found that if you export your spreadsheet into Excel the comments go over with it. If you then open a Google sheets and import the Excel spreadsheet, the comments come with it. Hopefully in the future Google will provide a choice “copy with comments” or “copy without comments.”

Although the comments look nicer when created in the original Google sheets, they are readable and workable when brought back from Excel.